Panel Data
Linear Models for Panel Data
This Block
We will do something
Learning Outcomes
References
- Chapter 17 in Huntington-Klein (2025): https://www.theeffectbook.net/ch-EventStudies.html
- Freyaldenhoven et al. (2021)
- Wooldridge (2020)
TYPES OF DATA
The Three Types of Data
Most (economic) research today uses panel
Why Panel Data
Key advantage of panel data: several observations of potential outcomes per unit
- Units with different treatments in different periods: (partially) observe the individual effect
- Units with no change in treatment: still can be used for learning about changes in other units
BINARY TREATMENT
EVENT STUDIES
Simple Event Study Setting
Begin with the simplest possible panel setting with binary treatment:
- Two periods with \(N\) units:
- No treatment in period 1.
- All units treated in period 2.
- Data: outcomes \((Y_{i1}, Y_{i2})\).
Object of interest: “average effect of treatment”
Simple Estimator — Average Change
Simplest approach: compute average change in \(Y_{it}\) across periods \[ \widehat{AE}_{ES} = \dfrac{1}{N}\sum_{i=1}^N (Y_{i2}- Y_{i1}). \tag{1}\]
Estimator (1) — simplest example of event study estimators (see Freyaldenhoven et al. 2021; Miller 2023).
Example Framework
Possible empirical framework
- Units \(i\): firms that make phones
- Outcome \(Y_{it}\): their stock price
- Periods:
- One week before Apple announces the iPhone.
- One week after the announcement
Effect of interest: change in stock prices due to the announcement of iPhone
What Does (1) Do?
Proposition 1 (Asymptotics for \(\widehat{AE}_{ES}\)) Let
- (Cross-sectional random sampling): \((Y_{i1}, Y_{i2})\) be independent and identically distributed (IID)
- Finite first moments: \(\E[\abs{Y_{it}}]<\infty\)
Then \[ \widehat{AE}_{ES} \xrightarrow{p} \E[Y_{i2} - Y_{i1}]. \]
Causal Framework
Is \(\E[Y_{i2} - Y_{i1}]\) interesting (=causal)?
Need a causal framework to talk about causal effects!
Work in the familiar potential outcomes framework:
- \(Y_{it}^0\) — outcome for \(i\) in period \(t\) if not treated
- \(Y_{i1}^1\) — outcome for \(i\) in period \(t\) if treated
- Treatment effect for \(i\) in \(t\): \(Y_{it}^1- Y_{it}^0\)
For short, use \(Y_{it}^d\) where \(d=0, 1\)
Limit of ES Estimator and Causality
Potential and realized outcomes are connected as \[ Y_{i2} = Y_{i2}^1, \quad Y_{i1} = Y_{i1}^0. \]
It follows that \[ \widehat{AE}_{ES} \xrightarrow{p} \E[Y_{i2}^2- Y_{i1}^1]. \]
\(\E[Y_{i2}^2- Y_{i1}^1]\) is not necessarily a treatment effect — mixes effect of treatment and effects of time!
Example
Context Again consider the iPhone example. Then
- \(Y_{i2}^1 - Y_{i2}^0\) — treatment effect, change in price because of the iPhone announcement
- \(Y_{i2}^0 - Y_{i0}^0\) — change in a world without iPhone
We see combination of both changes \[ Y_{i2} - Y_{i1} = Y_{i2}^1- Y_{i1}^0 = [Y_{i2}^1- Y_{i2}^0] + [Y_{i2}^0 - Y_{i0}^0] \]
Solution: Restrict Changes over Time
Simple solution: rule out changes over time
Assumption: no variation in potential outcomes \[ Y_{i2}^d= Y_{i1}^d, \quad d=0, 1 \]
Then \(\widehat{AE}_{ES}\) is estimating a causal parameter — average effects \[ \begin{aligned} \widehat{AE}_{ES} & \xrightarrow{p} \E[Y_{i1}^1- Y_{i1}^0] = \E[Y_{i2}^1- Y_{i2}^0] \end{aligned} \]
Relaxing Time Invariance: No Trends
Time invariance very strict. Why use it if we only work with averages?
Weaker assumption:
Assumption (no trends):
\[
\E[Y_{i2}^d] = \E[Y_{i1}^d], \quad d=0, 1
\]
Allows random variation in potential outcomes between times
Summary so Far
Proposition 2 (Causal asymptotics for \(\widehat{AE}_{ES}\)) Let
- Assumptions of Proposition 1 hold
- No trends assumption hold
Then \(\widehat{AE}_{ES}\) consistent for causal parameters: \[ \widehat{AE}_{ES} \xrightarrow{p} \E[Y_{i1}^1- Y_{i1}^0] = \E[Y_{i2}^1- Y_{i2}^0] \]
Regression Setting
Can also connect \(\widehat{AE}_{ES}\) and OLS
Consider regression model \[ \begin{aligned} Y_{it} & = \beta_0 + \beta_1 D_{it} + u_{it}, \\ D_{it} & = \begin{cases} 1, & t= 1 \\ 0, & t =0 \end{cases} \end{aligned} \tag{2}\] where we simply treat \((Y_{i1}, D_{i1})\) and \((Y_{i2}, D_{i2})\) as separate observations
Event Study and OLS
Proposition 3 (\(\widehat{AE}_{ES}\) is OLS) For \(\beta_1\) of regression (2) \[ \widehat{AE}_{ES} = \hat{\beta}_1^{OLS} \]
- Can use all results developed for OLS for \(\widehat{AE}_{ES}\)
- Regressing \(Y_{it}\) on \(D_{it}\) gives a causal parameter
- Under no trends
- No experiment was necessary
Event Study and Regression
A way to think about regression in causal settings:
Write down the regression in terms of parameters of interest: e.g. let \[ \beta_0 = \E[Y_{i0}^0], \quad \beta_1 = \E[Y_{i2}^1- Y_{i2}^0] \]
Connect regression to potential outcomes: what is \(u_{it}\) in terms of potential outcomes?
Check properties of this \(u_{it}\). If \(u_{it}\) is “nice”, apply OLS (or another method)
Extending to Multiple Periods and Adding Covariates
Expanded Regression
- Suppose \(T\) periods, treatment begins at \(t_0\).
- Regression model: \[ Y_{it} = \beta_0 + \sum_{\tau = t_0}^{T} \beta_\tau D_{it} + u_{it}. \]
- Estimates: \[ \hat{\beta}^{OLS} = \dfrac{1}{N} \sum_{i=1}^N (Y_{i\tau} - Y_{i1}). \]
Dynamic Treatment Effects
- No trends assumption rules out dynamic effects.
- Relax assumption: impose “no trends” on untreated outcome \(Y_{it}(0)\).
- Under this assumption: \[ \dfrac{1}{N} \sum_{i=1}^N (Y_{i\tau} - Y_{i1}) \xrightarrow{p} \E[Y_{i\tau}(1)- Y_{i1}(0)] = \E[Y_{i\tau}(1) - Y_{i\tau}(0)]. \]
- Consistently estimates average effects in period \(\tau\).
Event Study Origins
- Common in financial research.
- Cited in: (Freyaldenhoven et al. 2021; Miller 2023).
Pretrend Test
- Cannot directly test “no trends” assumption.
- Can examine pre-treatment period outcomes:
- If outcomes change over time, assumption might be problematic.
DiD
Slide
Things happen
LINEAR MODELS
SUBSECTION
Beyond Binary Treatments
We want
References
Panel Data